If I'm not misunderstanding anything, Jews usually follow the Tanakh (more or less the Old Testament) and the Talmud, but definitely not the New Testament, whereas Christians follow the Old and New Testaments but not the Talmud. So, among those who believe in the Old Testament, people usually follow the Talmud if and only if they do not believe in the New Testament.

Why is this? Are the Talmud and the New Testament incompatible, or is the combination simply unfashionable or something? Likewise, would a person ever follow the Old Testament, but neither the Talmud nor the New Testament?

4 Answers
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The Talmud "takes the form of a record of rabbinic discussions pertaining to Jewish law, ethics, philosophy, customs and history." (Wikipedia) It was written by ancient Rabbis as discussions and interpretations of the scriptures of the Tanakh (Old Testament).

The New Testament teaches that Scripture must be given by inspiration of God. As the Talmud was not, and does not claim to be, inspired writing, it is not considered authoritative by Christians.

Also, many of the passages of the Talmud are dedicated to interpretations of various provisions of the Law of Moses, which Christians believe were fulfilled in Christ and no longer apply, which makes a great deal of the Talmud not only non-authoritative but also irrelevant, from a Christian perspective.

Mason is right, but I would also add that in the Gospels, Jesus seems to treat the Tanakh (or in the Jewish idiom, "the Law and the Prophets") as Scripture and God's word.
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Patrick SzalapskiAug 31 '11 at 2:32

A short history of the Talmud

Jewish Law has it's foundation in the books of the Torah: the first five books of the Bible. Although there are something like 613 commandments (mitzvot) in the Torah, there are many areas of life that are not directly addressed by the written law. And society has changed in ways that aren't explicitly accounted for by the text. For instance, Moses commanded:

Mark that the Lord has given you the sabbath; therefore He gives you two days’ food on the sixth day. Let everyone remain where he is: let no one leave his place on the seventh day.—Exodus 16:29 (NJPS)

Originally, that mean to not leave the camp to go and gather manna. Traditionally it meant not to leave the area of the city as defined by the city walls. (See Jeremiah 17:19-27.) But these days, we don't have well-defined cities. I live near Los Angeles, but unless you pay careful attention to the signs on the side of the road, you can't tell if you are in Burbank, Glendale, Pasadena, Los Angeles or unincorporated Los Angeles county. Am I breaking mitzvot by moving around the metropolis? (The Orthodox Judaism solution is to create an eruv.)

To answer practical questions, such as this, Rabbis began to formulate a system of interpretations based on their understanding of the Mosaic principles. According to tradition, this Oral Torah could be traced back from student to teacher to Moses himself. After the destruction of the Second Temple, Rabbis began to write down the Oral Law, which became the Talmud.

Jesus and the Talmud

As it happens, during Jesus' time there were (according to Josephus and other historians) three mutually dissenting views on what constituted the Jewish Law:

Sadducees, who rejected not only the rabbinical traditions, but all of the Tanakh outside of the Torah.

Pharisees, who shaped and promoted the Oral Torah in addition to the Tanakh.

Essenes, who seemed to have accepted not only the Tanakh, but a wide range of other texts. They seem not to have accepted the Oral Torah. (If the Qumran community was Essene, we could be more certain about their beliefs.)

Jesus (and the early Christian movement) mostly interacted with the Pharisees and theologically seemed to have the most in common with them. (See Matthew 22:23-33 and parallels, Acts 5:17-42, and Acts 23:1-11.) Some scholars speculate that John the Baptist was an Essene, but otherwise we have no record of Jesus interacting with that sect. As you point out, Christianity accepts the texts that the Pharisees used as their scripture. (The Tanakh was completed and in the process of being canonized, while the Talmud remained strictly oral at the time.)

But Jesus roundly rejected the Oral Torah as misguided:

Woe to you, scribes and Pharisees, hypocrites! For you tithe mint and dill and cumin, and have neglected the weightier matters of the law: justice and mercy and faithfulness. These you ought to have done, without neglecting the others. You blind guides, straining out a gnat and swallowing a camel!—Matthew 23:23-24 (ESV)

But when Cephas came to Antioch, I opposed him to his face, because he stood condemned. For before certain men came from James, he was eating with the Gentiles; but when they came he drew back and separated himself, fearing the circumcision party. And the rest of the Jews acted hypocritically along with him, so that even Barnabas was led astray by their hypocrisy. But when I saw that their conduct was not in step with the truth of the gospel, I said to Cephas before them all, “If you, though a Jew, live like a Gentile and not like a Jew, how can you force the Gentiles to live like Jews?”—Galatians 2:11-14 (ESV)

The early church even rejected imposing certain laws written in the Torah, such as the kosher rules and circumcision, on gentile believers. As the church spread to Greek and Roman cities and beyond, the Jewish observance of the mitzvot also diminished. There is no evidence that Christianity ever advocated the Oral Torah which eventually became the Talmud.

Summary

Mason Wheeler summaried the Christian perspective on the the Talmud well. I will add that the Talmud provides a good deal of useful historical material for anyone trying to understand the origins of Christianity.

Jesus taught that we must follow the rabbinical (oral) law. Here is the verse:

Matthew 23:1-3
Then Jesus spoke to the multitudes and to his disciples, 2) saying,
"The scribes and the Pharisees sat on Moses' seat. 3) All things
therefore whatever they tell you to observe, observe and do"

Welcome to the site. I hope you don't mind, but I removed some unnecessary comments in your answer. One common misconception with newcomers is that this is a "discussion" site. It's not. Please check out the About page for a brief explanation of what this site is all about, and when you have more time, a trip to the FAQ and What Christianity.StackExchange is (and more importantly, what it isn't) will help you to be a more productive contributor. I do hope to see more of your answers in the future.
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David Stratton♦Jun 4 '13 at 12:31

Just a few verses later (Matt 23:15), Jesus says Woe to you, teachers of the law and Pharisees, you hypocrites! You travel over land and sea to win a single convert, and when you have succeeded, you make them twice as much a child of hell as you are. So, should we follow the laws but not the law-givers? How does that work?
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RichardJun 4 '13 at 12:33

During the time of Jesus Christ, Talmud was called Traditions of the Elders.

The Pharisees, the scribes, and all of the Jews who followed the sect of Pharisees nullified the word of God by obeying Traditions of the Elders (Matthew 15, Mark 7). Jesus calls Traditions of the Elders as human traditions (Mark 7:8) and severely criticized the Pharisees and the scribes for setting aside the commands of God in order to observe Traditions of the Elders which nullified the word of God (Mark 7, Matthew 15).

That is why the biggest opponents of Jesus Christ were the Pharisees and the scribes (Matthew 23, Luke 11). We also see Jesus Christ saying this - "Has not Moses given you the law? Yet not one of you keeps the law." (John 7:19). This is also mentioned by Stephen in Acts 7:53.

Traditions of the Elders was also known as Traditions of our Fathers (Galatians 1:14) or Traditions of our forefathers by Josephus in Antiquities of Jews.

"What I would now explain is this, that the Pharisees have delivered to the people a great many observances by succession from their fathers, which are not written in the laws of Moses; and for that reason it is that the Sadducees reject them, and say that we are to esteem those observances to be obligatory which are in the written word, but are not to observe what are derived from the tradition of our forefathers." (Antiquities of Jews Book XIII.X.VI).

This is also agreed by Jewish society.

Rabbi Michael Rodkinson - "The Talmud, then, is the written form of that which, in the time of Jesus, was called the Traditions of the Elders, and to which he makes frequent allusions" (Source - The History of the Talmud, Vol. II, page 70, Chapter IX).

So like Mason Wheeler pointed out above, the scripture must be God-breathed and should be useful for teaching, rebuking, correcting and training in righteousness, (2 Timothy 3:16)